Having had no less than fifteen visits from BT Openreach in the last ten months, I've heard so much conflicting b*llocks that I don't believe a word they tell me anymore.
After my problems started (slow bband, noisy 'phone and 'phone cutting out completely) I ran my router through a microfilter into the master socket for months up until it got changed for the iplate. I ran the 'phone to the same filter. There wasn't really any difference between master and normal socket, but did as I was told.
By the sound of it you have a fault in the rest of your 'phone wiring, which is eliminated when you disconnect it from the master. That fault could be miswiring, but also any strong interference near the wires to other 'phones. As an example, one engineer told me about an xbox near an extension socket that was causing all sorts of problems with the bband. The xbox was 'noisy' but still working.
I'd suggest unplugging everything else, then try the router speed connected to the face plate. If that's OK you can plug 'phones back in one by one, testing bband speed each time. If it's not OK, then the extension wiring may be damaged or collecting interference.
Mine still isn't fixed, btw!